# Generator of local symmetries

Let us only consider classical field theories in this discussion.

Noether's theorem states that for every global symmetry, there exists a conserved current and a conserved charge. The charge is the generator of the symmetry transformation. Concretely, if $\phi \to \phi + \varepsilon^a\delta_a \phi$ is a symmetry of the action, then there exists a set of conserved charges $Q_a$, such that $$\{Q_a, \phi\} = \delta_a \phi$$

If the symmetry is a gauge symmetry, then no such conserved current or charge exists (although certain constraints can be obtained via Noether's second theorem). However, I have the following question

Is there a quantity that, like $Q_a$ generates the gauge transformation?

This quantity need not be conserved. To be precise, if an action is invariant under local transformations $\phi \to \phi + [\varepsilon^a(x) \delta_a] \phi$. Here, heuristically, the gauge transformation might look like $[\varepsilon^a \delta_a] \phi \sim d \varepsilon + \varepsilon \phi + \cdots$ where $d \varepsilon$ is heuristically some derivative on $\varepsilon$. Does there exist a quantity such that $$\{ Q_{\varepsilon}, \phi \} = [ \varepsilon^a \delta_a] \phi$$ I know there exist such charges in some theories, and I have explicitly computed them as well. I was wondering if there is a general formalism to construct such quantities?

• Excuse me, I have a stupid question, how to show the Noether charge for local gauge transformation is zero? I looked for this topic in this website, but did not get the point ... Feb 20, 2014 at 2:20
• The second Noether states that the Noether current for any infinitesimal local symmetry vanishes on shell. However, this is based on the assumption that local symmetries act as the identity outside a bounded space-time region. Global gauge symmetries are not of this kind, they do not act as the identity as you move to infinity. Just think of Gauss's law: by measuring the electric flux through an arbitrarily large sphere, you can determine the amont of electric charge bounded by it. Feb 20, 2014 at 3:59
• For a thorough discussion on how Gauss's law is coherent with the second Noether theorem, I recommend the (long) paper by M. Forger and H. Römer, Currents and the Energy-Momentum Tensor in Classical Field Theory: A fresh look at an Old Problem, Annals Phys. 309 (2004) 306-389, arXiv:hep-th/0307199. See also the related physics.SE question physics.stackexchange.com/q/46476 Feb 20, 2014 at 4:09
• Doesn't the algebra of first class constraints generate the gauge transformations?
– Dan
Feb 20, 2014 at 4:44
• @Dan: That is correct (up to some pathological counter-examples), but the first class constraints are not Noether charges, they rather appear in the Noether identities that encode the on-shell vanishing of the corresponding Noether current. Feb 20, 2014 at 6:56

Let us here assume that the classical theory is given by a Hamiltonian (as opposed to a Lagrangian) formulation, so that we have a Poisson bracket $\{\cdot,\cdot\}_{PB}$ (and so that we can discuss whether or not the generators form a Poisson algebra or not).

A generic theory will have constraints (and corresponding Lagrange multipliers). The constraints are not a result of any Noether procedure, but are typically found systematically via the Dirac-Bergmann recipe when performing a (singular) Legendre transformation from the Lagrangian to the Hamiltonian formulation.

The constraints are divided into first and second-class constraints, say $G_a$, and $\chi_{\alpha}$, respectively. The Dirac conjecture states that the first-class constraints $G_a$ generate gauge symmetries

$$\delta_{\varepsilon}=\varepsilon^a\{G_a,\cdot\}_{PB},$$

cf. e.g. Ref. 1, which also lists a counterexample. Ref. 1 defines infinitesimal gauge transformations$^1$ $\delta_{\varepsilon}$ as quasisymmetry transformations that depend on infinitesimal gauge parameters $\varepsilon^a(x)$, which in turn are allowed to depend arbitrarily on the spacetime point $x$. Such spacetime-dependent (or so-called local) quasisymmetry transformations are the starting point of Noether's second theorem. See e.g. this Phys.SE post, where the corresponding Noether charges are discussed.

The presence of second-class constraints $\chi_{\alpha}$ imply that one must ultimately replace the Poisson bracket $\{\cdot,\cdot\}_{PB}$ with the pertinent Dirac bracket $\{\cdot,\cdot\}_{DB}$.

References:

1. M. Henneaux and C. Teitelboim, Quantization of Gauge Systems, 1994; Section 1.2.

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$^1$ Concerning the notion of gauge symmetry, see also e.g. this Phys.SE post.